Exact arithmetic and estimation · Arithmétique exacte et estimation
| English | Français |
|---|---|
| prime factor/praɪm ˈfæktə/ | facteur premier |
Can we pack without leftovers?
- A supplier packs 72 pencils and 90 pens into identical gift bags. How can we avoid leftovers?
- This lesson studies prime factor 质因数: A prime number that divides the integer exactly.
Choose the mathematical structure
- Prime factors reveal shared structure. Use the smallest common prime powers for the HCF and the largest for the LCM. Estimate before calculating; use brackets to preserve the order of operations.
- State the allowed inputs and units before calculating. An equation should express the relationship, not just record a calculator entry.
Which description correctly defines prime factor? · Quelle description définit correctement un facteur premier ?
A prime number that divides the integer exactly. · Un nombre premier qui divise l'entier exactement.
Work through a checked case
- Check the result against the starting quantities. Substitute into the original relation, or compare the graph and numerical answer where appropriate.
72=2^3×3^2 and 90=2×3^2×5. Their HCF is 2×9=18. Make 18 bags with 4 pencils and 5 pens each. Their LCM is 2^3×3^2×5=360.
Exact arithmetic and estimation · Arithmétique exacte et estimation
Prime factors reveal shared structure · Les facteurs premiers révèlent une structure commune
Combine the common powers 2 and 9 to justify 18 identical bags. · Combinez les puissances communes 2 et 9 pour justifier 18 sacs identiques.
Find the HCF of 48 and 60. · Trouver le PGCD de 48 et 60.
48=2⁴×3 and 60=2²×3×5. The shared smallest powers give HCF=2²×3=12. · 48=2⁴×3 et 60=2²×3×5. Les puissances communes les plus petites donnent PGCD=2²×3=12.
Test a tempting shortcut
- The HCF divides both numbers; the LCM is a multiple of both. They answer different questions. A decimal estimate is not an exact fraction.
- When a shortcut fails, identify the assumption it breaks. Keep an exact value until the requested final rounding.
The HCF of two positive integers is always larger than either integer. This claim is false. Explain which definition or assumption it violates.
Find the LCM of 8 and 12. · Trouver le PPCM de 8 et 12.
8=2³ and 12=2²×3. The largest prime powers give LCM=2³×3=24. · 8=2³ et 12=2²×3. Les puissances premières les plus grandes donnent PPCM=2³×3=24.
The HCF of two positive integers is always larger than either integer. · Le PGCD de deux entiers positifs est toujours supérieur à chacun des deux entiers.
The HCF divides both numbers; the LCM is a multiple of both. They answer different questions. A decimal estimate is not an exact fraction. · Le PGCD divise les deux nombres ; le PPCM est un multiple des deux. Ils répondent à des questions différentes. Une estimation décimale n'est pas une fraction exacte.
Interpret a new situation
- For a non-calculator paper, keep fractions exact and show cancellation. For a calculator paper, enter the full expression and compare with your estimate.
- A complete solution gives the mathematical result and explains what it means. Check that it is possible in the stated context.
Work out 3/4 + 5/6 as a decimal. · Calculer 3/4 + 5/6 en décimal.
Use denominator 12: 3/4+5/6=9/12+10/12=19/12. · Utiliser le dénominateur 12 : 3/4+5/6=9/12+10/12=19/12.
Match each part of a complete solution to its purpose. · Associer chaque partie d'une solution complète à son but.
An assumption justifies the model; a check tests the result; interpretation connects it to the question. · Une hypothèse justifie le modèle ; une vérification teste le résultat ; l'interprétation le relie à la question.
Use this in your course
- Current first-assessment-2021 Applications and Interpretation HL. This is authored concept support; the full guide is needed to certify every objective.
- Give the method before the final answer, and use the paper's calculator and formula rules. Review a wrong answer by locating the first invalid step.
A prime number that divides the integer exactly. Choose the relationship, show the method, check its assumptions and interpret the result.